On Using (Z^2, +) Homomorphisms to Generate Pairs of Coprime Integers

TL;DR

This paper introduces a novel method using homomorphisms on the group ^2 to generate all coprime positive integer pairs, analyzing their structure within a binary tree and exploring symmetry properties and conjectures on their ordering.

Contribution

It presents a new approach using ^2 homomorphisms to systematically generate coprime pairs and investigates their algebraic and combinatorial properties within a binary tree framework.

Findings

Sum of pairs is symmetric under reversal of homomorphism sequence

Provides a proof of symmetry for specific sequences of homomorphisms

Conjectures on the well-ordering of sums of coprime pairs

Abstract

We use the group $(\Z^2,+)$ and two associated homomorphisms, $\tau_0, \tau_1$, to generate all distinct, non-zero pairs of coprime, positive integers which we describe within the context of a binary tree which we denote $T$. While this idea is related to the Stern-Brocot tree and the map of relatively prime pairs, the parents of an integer pair these trees do not necessarily correspond to the parents of the same integer pair in $T$. Our main result is a proof that for $x_i \in \{0,1\}$, the sum of the pair $\tau_{x_1}\tau_{x_2}... \tau_{x_n} [1,2]$ is equal to the sum of the pair $\tau_{x_n}\tau_{x_{n-1}} ... \tau_{x_1} [1,2]$. Further, we give a conjecture as to the well-ordering of the sums of these integers.